Theorem sltssn 27921

Description: Surreal set less-than of two singletons. (Contributed by Scott Fenton, 17-Mar-2025.)

Hypotheses

Ref	Expression
sltssn.1	⊢ (𝜑 → 𝐴 ∈ No )
sltssn.2	⊢ (𝜑 → 𝐵 ∈ No )
sltssn.3	⊢ (𝜑 → 𝐴 <s 𝐵)

Assertion

Ref	Expression
sltssn	⊢ (𝜑 → {𝐴} <<s {𝐵})

Proof of Theorem sltssn

Step	Hyp	Ref	Expression
1		sltssn.3	. 2 ⊢ (𝜑 → 𝐴 <s 𝐵)
2		sltssn.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
3		sltssn.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
4	2, 3	sltssnb 27920	. 2 ⊢ (𝜑 → ({𝐴} <<s {𝐵} ↔ 𝐴 <s 𝐵))
5	1, 4	mpbird 260	1 ⊢ (𝜑 → {𝐴} <<s {𝐵})

Syntax hints:   → wi 4   ∈ wcel 2145  {csn 4585   class class class wbr 5105   No csur 27762   <s clts 27763   <<s cslts 27908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-slts 27909

This theorem is referenced by:  cutneg  27967  zcuts  28558  twocut  28574  nohalf  28575  pw2recs  28589  halfcut  28609  addhalfcut  28610  pw2cut2  28613  bdaypw2n0bndlem  28614  bdayfinbndlem1  28618